====== Kelvin Lee ======

===== Personal Notes =====

Here is the link to my personal course notes for this class. [[https://kelvinlee200113.github.io/notes/104.pdf|Notes]] (They might contain typos or logical errors.)
They are created based on Ross's, Rudin's textbooks and Professor Zhou's lectures.


===== Questions =====

1. What's the difference between continuity and uniform continuity ? \\
2. What's the difference between pointwise convergence and uniform convergence? \\
3. Is a series of continuous functions necessarily continuous? \\
** Answer: ** No, consider $f_n(x)=x^{n}-x^{n-1}$ and $\sum_n f_n(x)$ is not continuous.\\
4. (Yuwei Fan's practice final) Let $f:[a, b] \rightarrow \mathbb{R}$ be an integrable function. Prove that
$$
\lim _{n \rightarrow \infty} \int_{a}^{b} f(x) \sin (n x) d x=0.
$$
5. (Yuwei Fan's practice final) Suppose that $f:[1, \infty)$ $\rightarrow \mathbb{R}$ is \text{uniformly continuous} on $[1, \infty)$. Prove that there exists $M>0$ such that $\frac{|f(x)|}{x} \leq M$ holds for any $x \geq 1$. \\
6. What are some nice properties that continuity preserves? \\
** Answer: ** Compactness, connectedness.
7. What does it mean intuitively for a set to be both closed and open?\\
8. What is the motivation behind the concept of compactness? \\
9. If $\{f_n\}$ are continuous, does it mean that its limit is also continuous? \\
10. (Yuwei Fan's practice final) For a bounded function $f:[0,1] \rightarrow \mathbb{R}$, define
$$
R_{n}:=\frac{1}{n} \sum_{k=1}^{n} f\left(\frac{k}{n}\right)
$$
(a) Prove that if $f$ is integrable, then $\lim _{n \rightarrow \infty} R_{n}=\int_{0}^{1} f(x) d x$.\\
(b) Find an example of $f$ that is not integrable, but $\lim _{n \rightarrow \infty} R_{n}$ exists.\\
11. If $f_{n} \rightarrow f$ uniformly on $S$, then $f_{n}^{\prime} \rightarrow f^{\prime}$ uniformly on $S$. \\
** Answer **: False.\\
12. If $f$ is differentiable on $[a, b]$ then it is integrable on $[a, b]$.\\
** Answer **: True.\\
13. If $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous on $[a, b]$, there is a sequence of polynomials whose uniform limit on $[a, b]$ is $f .$ \\
** Answer **: True.\\
14. Let $f$ and $g$ be continuous functions on $[a, b]$ such that $\int_{a}^{b} f=\int_{a}^{b} g$. Show that there is an $x \in[a, b]$ such that $f(x)=g(x)$. \\
15. Let $\left\{f_{n}\right\}$ be a sequence of continuous functions on $[a, b]$ that converges uniformly to $f$ on $[a, b] .$ Show that if $\left\{x_{n}\right\}$ is a sequence in $[a, b]$ and if $x_{n} \rightarrow x$, then $\lim _{n \rightarrow \infty} f_{n}\left(x_{n}\right)=f(x)$. \\
16. Find an example or prove that the following does not exist: a monotone sequence
that has no limit in $\mathbb{R}$ but has a subsequence converging to a real number. \\
17. Consider a continuous function $f$ on $(0, \infty)$, and suppose that $f$ is a uniformly continuous on $(0, a)$ for all $a>0$. Then $f$ must be a uniformly continuous function on $(0, \infty)$. \\
18. Consider a sequence $\left(f_{n}\right)_{n=1}^{\infty}$ of continuous functions on $[0,1]$. Suppose that $\left(f_{n}\right)$ converges pointwise to a function $f$ on $[0,1]$, and that
$$
\lim _{n \rightarrow \infty} \int_{0}^{1} f_{n}(x) d x=\int_{0}^{1} f(x) d x
$$
Then, $\left(f_{n}\right)$ must converge to $f$ uniformly on $[0,1]$. \\
19. Suppose that a sequence of functions $\left(f_{n}\right)_{n=1}^{\infty}$ converges to $f$ uniformly on $(0,1)$. Then, the sequence $\left(f_{n}^{3}\right)_{n=1}^{\infty}$ converges to $f^{3}$ uniformly on $(0,1)$. \\
20. Let $0<a<1$ be a fixed number. Suppose that a sequence of functions $\left(g_{n}\right)_{n=1}^{\infty}$ on $[0, a]$ satisfies $\left|g_{n}(x)\right| \leq x^{n}$ for all $x \in[0, a]$ and for all $n \in \mathbb{N}$. Then, $\sum g_{n}(x)$ is a uniformly convergent infinite series. \\
21. Suppose that a sequence of functions $\left(f_{n}\right)_{n=1}^{\infty}$ on $[0,1]$ converges uniformly to $f$ on $[0,1]$. Let $g$ be a continuous function on $[0,1]$. Prove that $\left(f_{n} g\right)_{n=1}^{\infty}$ converges uniformly to $f g$ on $[0,1]$. \\
22. $\left(10\right.$ points) Let $\sum a_{n}$ be a convergent series and $\left(f_{n}\right)$ be a sequence of real-valued functions defined on $S \subset \mathbb{R}$ such that
$$
\left|f_{n+1}(x)-f_{n}(x)\right|<a_{n}, \quad \forall n \in \mathbb{N}, \forall x \in S
$$
Prove that $\left(f_{n}\right)$ is uniformly Cauchy on $S$ and hence it is uniformly convergent on $S$.\\
23. Let $\alpha$ be a bounded, monotonically increasing function on $\mathbb{R}$. What is
$$
\int_{-\infty}^{\infty} 1 d \alpha ?
$$
24. Suppose $f$ is continuous on $[a, b]$ and $\alpha$ is continuous and strictly increasing. Show that if
$$
\int_{a}^{b} f^{2}(x) d \alpha=0
$$
then $f$ is identically 0 on $[a, b]$.\\
25. Suppose that $f$ and $g$ are continuous functions on $[a, b]$ such that $\int_{a}^{b} f=\int_{a}^{b} g .$ Prove that there exists $x \in[a, b]$ such that $f(x)=g(x)$. \\
26. Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function and that $f^{\prime}(x)$ exists and is bounded on $\mathbb{R}$. Show that $f$ is uniformly continuous on $\mathbb{R}$. \\
27. Let $f$ be a (Darboux) integrable function on $[a, b]$ and $F$ a differentiable function on $[a, b]$ with $F^{\prime}(x)=f(x)$ except for finitely many $x \in[a, b] .$ Show that $f^{\prime \prime}$ is integrable as well and conclude:
$$
\int_{a}^{b} f=F(b)-F(a).
$$
28. Give a complete proof of the integral criterion for convergence of series. Namely for a monotone decreasing function $f:[0, \infty) \longrightarrow \mathbb{R}$ with $f(x) \geq 0$ for all $x \geq 0$ and
$$
\lim _{b \rightarrow \infty} \int_{0}^{b} f<\infty
$$
the series $\sum_{m=1}^{\infty} f(m)$ converges (absolutely).\\
29. If $f(x)$ and $g(x)$ are uniformly continuous on $\mathbb{R}$, then $f \cdot g$ is uniformly continuous on $\mathbb{R}$. \\
** Answer **: False.\\
30. $(10$ points) A function $f: \mathbb{R} \rightarrow \mathbb{R}$ is called periodic if there exists a $T>0$ such that $f(x)=f(x+T)$ for all $x \in \mathbb{R} .$ Suppose that $f$ is a periodic function that is differentiable on $\mathbb{R}$. Show that there exists $x \in \mathbb{R}$ such that $f^{\prime}(x)=0$. \\
31. Let $[a, b]$ be an interval and $c \in(a, b)$. Define a function $f$ on $[a, b]$ as follows: \\ 
$$ f(x)=\begin{cases} 0 & x \neq c \\ 1 & x=c \end{cases}.$$
Show that $f$ is integrable on $[a, b]$, and find $\int_{a}^{b} f$. \\
32. Let $g$ be an integrable function on $[a, b]$ and suppose that $h(x)=g(x)$ for all but one $x$ in $[a, b] .$ Show that $h$ is integrable and that $\int_{a}^{b} g=\int_{a}^{b} h$. \\
33. Consider $f:[a, b] \rightarrow \mathbb{R}$. Suppose that $f^{\prime}$ is bounded where it exists (do not assume it exists everywhere). Then, $f$ is bounded.\\
34. 7. Let $f$ be an infinitely differentiable function on $\mathbb{R}$. Then the Taylor series of $f$ at any point $a \in \mathbb{R}$ converges to $f$ in some neighborhood of $a$. \\
35. Define
$$
f_{n}(x)=\sum_{k=1}^{n} \frac{1}{k^{2} x^{2}}
$$
defined on $(0, \infty)$. The $f_{n}$ converge uniformly.\\
36. Suppose $a_{n}>0$ and $\sum a_{n}$ converges. Then $\sum \frac{a_{n}+a_{n+1}}{2}$ converges.\\
37. A function $f: \mathbb{R} \rightarrow \mathbb{R}$ is said to satisfy the Lipshitz condition of order $\alpha$ at $a \in \mathbb{R}$ if there is a constant $M$ and a neighborhood of $a$ such that
$$
\left|f(x)-f(a)\right|<M|x-a|^{\alpha}
$$
Show that if $f$ has the Lipschitz condition of order $\alpha$ for $\alpha>0$, then $f$ is continuous. Give an example of a function $f$ which has Lipschitz condition of order 0 which is not continuous at $c$. \\
38. Show that the function $f(x)=x^{a}$ is uniformly continuous for $a>1$.\\
39. If $f: X \rightarrow Y$ is continuous and is not a constant function (i.e. $f(X)$ has more than one point), and $y \in Y$ is an isolated point, then $f^{-1}(\{y\})$ consists only of isolated points. \\
40. Let $E \subset \mathbb{R}$ be a closed subset. There is some $F \subset \mathbb{R}$ whose set of limit points is exactly $E$. (Note that isolated points of $E$ are not limit points of $E$.) \\
41. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be continuous. If a sequence $p_{n}$ in $\mathbb{R}$ diverges, then $f\left(p_{n}\right)$ also diverges. \\
42. Let $f$ be a function defined on $(0,1]$, and suppose $f$ is integrable on every interval $[c, 1]$ for $c \in(0,1)$. Define:
$$
\int_{0}^{1} f d x=\lim _{c \rightarrow 0} \int_{c}^{1} f d x
$$
Show that if $f$ is defined on $[0,1]$ and is integrable, then this definition agrees with the usual one. Find an example of a function $f:(0,1] \rightarrow \mathbb{R}$ such that the above integral exists for $f$ but not for $|f|$. \\
43. Suppose $f$ is bounded and real, and suppose $f^{2}$ is Riemann integrable (with respect to $\left.\alpha(x)=x\right)$. Does it follow that $f$ is Riemann-integrable? \\
44. Suppose $f$ is bounded and real, and suppose $f^{3}$ is Riemann integrable (with respect to $\left.\alpha(x)=x\right)$. Does it follow that $f$ is Riemann-integrable? \\
45.Suppose that $f$ is infinitely differentiable everywhere. Suppose that there is some $L>0$ such that $\left|f^{(n)}(x)\right|<L$ for all $n$ and $x \in \mathbb{R}.$ Further suppose that $f\left(\frac{1}{n}\right)=0$ for all $n \in \mathbb{N} .$ Show that $f(x)=0$ everywhere. \\
46. Define the integral:
$$
\int_{a}^{\infty} f d x:=\lim _{b \rightarrow \infty} \int_{a}^{b} f d x
$$
If it exists, we say it converges. Suppose that $f(x) \geqslant 0$ and $f$ decreases monotonically on $[1, \infty)$. Show that
$$
\int_{1}^{\infty} f d x
$$
converges if and only if
$$
\sum_{n=1}^{\infty} f(n)
$$
converges.\\
47. Suppose $\alpha$ is monotonically increasing and continuous on $[a, b]$ and $p \in[a, b] .$ Define $f$ by $f(p)=1$ and $f(x)=0$ for $x \neq p$. Show directly (do not cite a theorem other than the basic "Cauchy criterion" for integrability) that $f$ is Riemann integrable with respect to $\alpha$ and that $\int_{a}^{b} f d \alpha=0$. \\
48. Prove that
$$
\sum_{n=2}^{\infty} \frac{1}{n(\log (n))^{p}}
$$
converges for $p>1$ and diverges for $p \leqslant 1$.\\
49. Prove that if $a_{n}$ is a decreasing sequence of real numbers and if $\sum a_{n}$ converges, then $\lim n a_{n}=0$. \\
50. Let $f$ and $g$ be continuous functions on $[a, b]$ that are differentiable on $(a, b)$. Suppose that $f(a)=f(b)=0 .$ Prove that there exists $x \in(a, b)$ such that
$g^{\prime}(x) f(x)+f^{\prime}(x)=0$.